The correct reason or ground for the British Empire to annex Sambalpur in 1850 was the death of its ruler without any heir.

Option 1: Lack of proper governance is not mentioned as the reason for annexation. This option suggests that Sambalpur was annexed due to a lack of effective administration, but there is no information provided to support this.

Option 3: Subsidiary Alliance is not the reason for annexation in this case. Subsidiary Alliance was a policy introduced by the British to establish control over princely states in India. However, there is no indication that Sambalpur was annexed through this policy.

Option 4: Its involvement in a conspiracy against British Rule is not mentioned as a reason for annexation. This option implies that Sambalpur was annexed because it was involved in a plot against the British, but there is no evidence to support this claim.

Therefore, the correct answer is option 2: Death of its ruler without any heir. This indicates that the British Empire took advantage of the absence of a successor to the throne and used it as an opportunity to annex Sambalpur.